Big subsets with small boundaries in a graph with a vertex-transitive group of automorphisms

The theory of ends of finitely generated groups $G$ and connected locally finite graphs $\Gamma$ with vertex-transitive groups of automorphisms can be regarded as a theory of Boolean algebras of subsets of $G$ or vertex set of $\Gamma$ with finite boundaries (in the locally finite Cayley graph of $G$ or in $\Gamma$ respectively), considered modulo finite subsets. We develop a more general theory where infinite subsets with finite boundaries are replaced by certain ‘big’ subsets with ‘small’ boundaries.